Project description: 

We propose to develop a novel mathematical and computational approach for data-intensive nonlinear filtering problems. The techniques of linear filtering have contributed tremendously in simulating dynamical systems, but they are only first-order approximations to nonlinear filtering problems. Our objective is to significantly improve the applicability and efficiency of nonlinear filtering simulations by exploring substantially novel directions based on the theory of the equivalence between the nonlinear filtering problems and a class of backward stochastic differential equations (BSDEs). In our methodology, a nonlinear filtering problem is handled by numerically solving a BSDE, in the sense of which several fundamental challenges, e.g. massive data, high dimensionality and non-Gaussian noise, etc., can be addressed . In addition, a truly scalable filtering capability will be established for applications of importance to the DOE mission, and the proposed algorithms will be made available through a recently developed ORNL Toolkit for Adaptive Stochastic Modeling and Non-Intrusive ApproximatioN (TASMANIAN).

Sponsor: ORNL - Laboratory Directed Research & Development

Funding period: 2014 -- 2015

Publications:

1. H. Tran, C. Webster and G. Zhang, A sparse grid method for Bayesian uncertainty quantification with application to large eddy simulation turbulence models, Springer Lecture Notes on CS&E, 109 (2016), pp. 291-313. 

2. F. Bao, Y. Cao, C. Webster and G. Zhang, An efficient meshfree implicit filter for nonlinear filtering problems, International Journal for Uncertainty Quantification, 6 (2016), pp. 19-33. 

3. G. Zhang, W. Zhao, M. Gunzburger and C. Webster, Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs, Computers & Mathematics with Applications, 71 (2016), pp. 2479-2496. 

4. G. Zhang, C. Webster, M. Gunzburger and J. Burkardt, A hyper-spherical adaptive sparse-grid method for high-dimensional discontinuity detection, SIAM Journal of Numerical Analysis, 53 (2015), pp. 1508-1536. 

5. V. Reshniak, A. Khaliq, D. Voss and G. Zhang, Split-step Milstein methods for multi-channel stiff stochastic differential systems, Applied Numerical Mathematics, 89 (2015), pp. 1-23. 

6. F. Bao, Y. Cao, C. Webster and G. Zhang, A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive domain of the Zakai equation approximations, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), pp. 784-804. 

7. C. Webster, G. Zhang and M. Gunzburger, An adaptive sparse-grid-based iterative ensemble Kalman filter approach for parameter field estimation, International Journal of Computer Mathematics, 91 (2014), pp. 798-817. 

Invited talks:

• 13th US National Congress on Computational Mechanics, San Diego, CA (July 2015)

• 45th Annual John H. Barrett Memorial Lectures, University of Tennessee, Knoxville, TN (May 2015)

• SIAM Conference on Computational Science and Engineering, Salt Lake City, UT (Mar. 2015)

• The 9th International Conference on Computational Physics, Singapore (Jan. 2015)

• American Geophysics Union Annual Meeting, San Francisco, CA (Dec. 2014)

• Department of Mathematics and Statistics, Auburn University, Auburn, AL (Nov. 2014)

• Department of Mathematics, University of Tennessee, Knoxville, TN (Oct. 2014)

• 3rd Workshop on Sparse Grid and Applications, Stuttgart, Germany (Sep. 2014)

Other activities:

• Guannan co-organized a mini-symposium on "High-dimensional Approximation and Integration: Analysis and Computation", at SIAM  Conference on Computational Science and Engineering (Mar. 2015) 

• Guannan co-organized a mini-symposium on "Theoretical and numerical analysis for forward-backward stochastic differential equations and stochastic optimal control", at 2014 SIAM  Conference on Uncertainty Quantification (Apr. 2014)